Type-I Multiferroic VHfO${}_4$ with Strain-Switchable Magnetic Orders and Magnetoelectric Coupling

Motivated by the complementary properties of vanadium-based ferromagnets and HfO${}_2$-based ferroelectrics, we propose a novel multiferroic oxide, VHfO${}_4$, through 50% Hf${}^{4+}$ substitution with V${}^{4+}$ in the ferroelectric $Pca{2}_1$ phase of HfO${}_2$. First-principles DFT calculations reveal that the $Pca{2}_1$-like VHfO${}_4$ phase exhibits dynamic stability and concurrent ferroic orders: robust ferroelectric polarization comparable to HfO${}_2$ and V-driven magnetism. Parallel tempering Monte Carlo simulations identify an antiferromagnetic ground state, while strain engineering enables tunable magnetoelectric coupling. Biaxial in-plane strain induces four magnetic states: intralayer FM/interlayer AFM, intralayer AFM/interlayer FM, spiral-like non-collinear order, and discrete alternating spin alignment. Critically, $c$-axis strain modulates magnetic energy landscapes, demonstrating electromechanical control of magnetism. This work establishes VHfO${}_4$ as a Type-I multiferroic with coexisting atomic-scale ferroic origins and strain-tunable cross-coupling, offering a platform for voltage-controlled spintronics devices.